John Baez

Public Lecture at Categories in Algebra, Geometry and Mathematical Physics

July 14, 2005

The Mysteries of Counting:
Euler Characteristic versus Homotopy Cardinality

The Mysteries of Counting - transparencies in PDF format.

For more information, try these papers:

• "Negative sets" and Euler characteristic:
  • André Joyal, Regle des signes en algebre combinatoire, Comptes Rendus Mathematiques de l'Academie des Sciences, La Societe Royale du Canada VII (1985), 285-290.

  • Steve Schanuel, Negative sets have Euler characteristic and dimension, Lecture Notes in Mathematics 1488, Springer Verlag, Berlin, 1991, pp. 379-385.

  • Daniel Loeb, Sets with a negative number of elements, Adv. Math. 91 (1992), 64-74.

• Resumming divergent Euler characteristics:
  • William J. Floyd and Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1-29.

  • R. I. Grigorchuk, Growth functions, rewriting systems and Euler characteristic, Mat. Zametki 58 (1995), 653-668, 798.

  • James Propp, Exponentiation and Euler measure, Algebra Universalis 49 (2003), 459-471. Also available as math.CO/0204009.

  • James Propp, Euler measure as generalized cardinality. Available as math.CO/0203289.

• Euler characteristics and supersymmetry:
  • Matthias Blau and George Thompson, N = 2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant, Comm. Math. Phys. 152 (1993), 41-71.

• Euler characteristics of tame spaces:

  • Lou van den Dries, Tame Topology and O-Minimal Structures Cambridge U. Press, Cambridge, 1998. Chapter 4.2: Euler Characteristic.

• Euler characteristics of groups:

  • G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 409-455.

  • Jean-Pierre Serre, Cohomologie des groups discretes, Ann. Math. Studies 70 (1971), 77-169.

  • Kenneth S. Brown, Euler characteristics, in Cohomology of Groups, Graduate Texts in Mathematics 182, Springer, 1982, pp. 230-272.

• Euler characteristics of categories and n-categories:
  • Tom Leinster, The Euler characteristic of a category. Available as arXiv:math/0610260.

  • Tom Leinster, The Euler characteristic of a category as a sum of a divergent series. Available as arXiv:0707.0835.

• Euler characteristics of chain complexes of graded vector spaces:
  • Igor Frenkel, Catharina Stroppel, and Joshua Sussan, Categorifying fractional Euler characteristics, the Jones-Wenzl projector and 3j-symbols.

• Bijective proofs in combinatorics:
  • James Propp and David Feldman, Producing new bijections from old, Adv. Math. 113 (1995), 1-44. Also available at: http://faculty.uml.edu/jpropp/cancel.pdf

  • John Horton Conway and Peter G. Doyle, Division by three. Available at: https://arxiv.org/abs/math/0605779

• Complex cardinalities:

  • Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg. 103 (1995), 1-21. Also available at: http://www.math.lsa.umich.edu/~ablass/cat.html

  • Robbie Gates, On the generic solution to P(X) = X in distributive categories, Jour. Pure Appl. Alg. 125 (1998), 191-212.

  • Marcelo Fiore and Tom Leinster, An objective representation of the Gaussian integers, Jour. Symb. Comp. 37 (2004), 707-716. Also available as math/0211454.

  • Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, Adv. Math. 190 (2005), 264-277. Also available as math/0212377.

  • Marcelo Fiore, Isomorphisms of generic recursive polynomial types, to appear in 31st Symposium on Principles of Programming Languages (POPL04). Also available at http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz

• Generalized sets with nonnegative real cardinalities:
  • George Janelidze and Ross Street, Real sets, Tbilisi Mathematical Journal 10 (2017), 23-49. Corrected version available as arXiv:1704.08787.

• Applications to quantum theory:
  • John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50. Also available as math/0004133.

  • Jeffrey Morton, Categorified algebra and quantum mechanics, TAC 16 (2006), 785-854.

• Week 147 - Categorification, Euler characteristic versus homotopy cardinality, and the cardinality of the fundamental group of a Riemann surface.

• Week 184 - q-deformed cardinalities: how cardinalities of Grassmannians over finite fields relate to Euler characteristics of Grassmannians over R and C.

• Week 185 - Vector spaces over the field with q elements as a q-deformation of finite sets; more on q-deformed cardinalities.

• Week 186 - The q-polynomial of a simple algebraic group, and using it to compute the cardinalities of flag manifolds over finite fields, and their Euler characteristics over R and C.

• Week 187 - The q-polynomial of a simple algebraic group - the classical groups treated in detail.

• Week 202 - Joyal's species and reasoning with complex cardinalities.

• Week 203 - An object whose cardinality is the golden ratio.

• Week 244 - Leinster's work on the Euler characteristic of a category.

• John Baez and Derek Wise, Quantization and Categorification.
  
  • Fall 2003 notes: http://math.ucr.edu/home/baez/qg-fall2003
    

  • Winter 2004 notes: http://math.ucr.edu/home/baez/qg-winter2004/
    

  • Spring 2004 notes: http://math.ucr.edu/home/baez/qg-spring2004/

• John Baez, Euler characteristic versus homotopy cardinality, lecture at the Fields Institute Program on Applied Homotopy Theory, September 20, 2003. Available in PDF form at: http://www.math.ucr.edu/home/baez/cardinality/

Mathematics is not the rigid and rigidity-producing schema that the layman thinks it is; rather, in it we find ourselves at that meeting point of constraint and freedom that is the very essence of human nature. - Hermann Weyl

© 2005 John Baez
baez@math.removethis.ucr.andthis.edu

home